Problem: Simplify the following expression and state the condition under which the simplification is valid: $k = \dfrac{x^2 - 7x + 6}{x^2 - 6x}$
First factor the expressions in the numerator and denominator. $ \dfrac{x^2 - 7x + 6}{x^2 - 6x} = \dfrac{(x - 1)(x - 6)}{(x)(x - 6)} $ Notice that the term $(x - 6)$ appears in both the numerator and denominator. Dividing both the numerator and denominator by $(x - 6)$ gives: $k = \dfrac{x - 1}{x}$ Since we divided by $(x - 6)$, $x \neq 6$. $k = \dfrac{x - 1}{x}; \space x \neq 6$